Bootstrapping the Three Dimensional Supersymmetric Ising Model

We implement the conformal bootstrap program for three dimensional conformal field theories with $\mathcal{N}=2$ supersymmetry and find universal constraints on the spectrum of operator dimensions in these theories. By studying the bounds on the dimension of the first scalar appearing in the operator product expansion of a chiral and an antichiral primary, we find a kink at the expected location of the critical three dimensional $\mathcal{N}=2$ Wess-Zumino model, which can be thought of as a supersymmetric analog of the critical Ising model. Focusing on this kink, we determine, to high accuracy, the low-lying spectrum of operator dimensions of the theory, as well as the stress-tensor two-point function. We find that the latter is in an excellent agreement with an exact computation.

Figure 1

Bound on the dimensions of the leading unprotected operator in the $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ OPE at $n_{\max }=9$. There can be no unitary SCFTs in the white region. ${\mathrm{\Delta }}_{\mathrm{\Phi }}=2/3$ is indicated with a red dashed line. The small shaded rectangle at ${\mathrm{\Delta }}_{\mathrm{\Phi }}=2/3$ indicates the field of view in Fig. 2.

Figure 2

Bound on the dimension of the leading unprotected operator in the $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ OPE close to ${\mathrm{\Delta }}_{\mathrm{\Phi }}=2/3$ (upper curves), and the corresponding central charge $C_T$ of the solution on the boundary (lower curves). The numbers in parenthesis next to the values of $n_{\max }$ indicate the number of constraints imposed to generate the associated curve.

Figure 3

A closer view of Fig. 2. The shaded rectangles indicate our estimated error for ${\mathrm{\Delta }}_{[\mathrm{\Phi }\overline{\mathrm{\Phi }}]}$ and $C_T$.

Figure 4

Bound on the dimension of the subleading superconformal scalar primary, $[\mathrm{\Phi }\overline{\mathrm{\Phi }}{]}^{\prime }$, in the $\mathrm{\Phi }\times \overline{\mathrm{\Phi }}$ OPE. The dimension is extracted from the solution that maximizes ${\mathrm{\Delta }}_{[\mathrm{\Phi }\overline{\mathrm{\Phi }}]}$.

Figure 5

Charged scalar spectrum in the vicinity of ${\mathrm{\Delta }}_{\mathrm{\Phi }}=2/3$. Left: The first three spin zero operators in the spectrum (colored by order of appearance). The dashed lines correspond to $2{\mathrm{\Delta }}_{\mathrm{\Phi }}$, $d-2{\mathrm{\Delta }}_{\mathrm{\Phi }}$, and $2(d-1)-2{\mathrm{\Delta }}_{\mathrm{\Phi }}$ with $d=3$ (see [20]). Right: The OPE coefficients for each operator appearing on the left-hand plot (with matching colors). Observe the vanishing of the ${\mathrm{\Phi }}^2$ OPE coefficient at ${\mathrm{\Delta }}_{\mathrm{\Phi }}\simeq 2/3$. The “noisy” operators in the spectrum plots can be seen to have vanishingly small OPE coefficients and thus correspond to small numerical artifacts in the solution.

Figure 6

The spin-1 spectrum for $n_{\max }=10$ in the vicinity of ${\mathrm{\Delta }}_{\mathrm{\Phi }}=2/3$. The decoupling of the subleading spin-1 field is reminiscent of what happens in the spin-2 sector of the nonsupersymmetric Ising model [22]. In the inset we show the spin-1 spectrum for various values of $n_{\max }$ which all lie well within our estimated error bars.